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Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 2 February 2008 (MN LATEX style file v2.2)

Considerations in optimizing CMB polarization

experiments to constrain inflationary physics

Licia Verde1⋆, Hiranya V. Peiris2† and Raul Jimenez1‡1Dept. of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA-19104, USA.2Kavli Institute for Cosmological Physics and Enrico Fermi Institute, University of Chicago, Chicago IL 60637, USA.

2 February 2008

ABSTRACT

We quantify the limiting factors in optimizing current-technology cosmic mi-crowave background (CMB) polarization experiments to learn about inflationaryphysics. We consider space-based, balloon-borne and ground-based experiments. Wefind that foreground contamination and residuals from foreground subtraction are ul-timately the limiting factors in detecting a primordial gravity wave signal. For full-skyspace-based experiments, these factors hinder the detection of tensor-to-scalar ratiosof r < 10−3 on large scales, while for ground-based experiments these factors impedethe ability to apply delensing techniques. We consider ground-based/balloon-borneexperiments of currently planned or proposed designs and find that it is possible fora value of r = 0.01 to be measured at ∼ 3-σ level. A small space-based CMB polar-ization experiment, with current detector technology and full sky coverage, can detectr ∼ 1 × 10−3 at the ∼ 3-σ level, but a markedly improved knowledge of polarizedforegrounds is needed. We advocate using as wide a frequency coverage as possible inorder to carry out foreground subtraction at the percent level, which is necessary tomeasure such a small primordial tensor amplitude.

To produce a clearly detectable (>3-σ) tensor component in a realistic CMB exper-iment, inflation must either involve large-field variations, ∆φ >

∼ 1 or multi-field/hybridmodels. Hybrid models can be easily distinguished from large-field models due to theirblue scalar spectral index. Therefore, an observation of a tensor/scalar ratio and n < 1in future experiments with the characteristics considered here may be an indicationthat inflation is being driven by some physics in which the inflaton cannot be describedas a fundamental field.

Key words: cosmology: cosmic microwave background — cosmology: theory —cosmology: early universe

1 INTRODUCTION

The inflationary paradigm, proposed and elucidated byGuth (1981); Linde (1982); Albrecht & Steinhardt (1982);Sato (1981); Mukhanov & Chibisov (1981); Hawking (1982);Guth & Pi (1982); Starobinsky (1982); Bardeen et al.(1983); Mukhanov et al. (1992) has enjoyed resounding suc-cess in the current era of precision cosmology. Not only hasits broad-brush prediction of a flat universe been success-ful, but the more detailed predictions of the simplest in-flationary models, of Gaussian, adiabatic, and nearly (butnot exactly) scale-invariant perturbations, are also perfectly

⋆ lverde@physics.upenn.edu† Hubble Fellow, hiranya@cfcp.uchicago.edu‡ raulj@physics.upenn.edu

consistent with the data. Observations of the cosmic mi-crowave background (CMB), in particular by the WMAPsatellite (see Bennett et al. (2003a) and references therein),have been instrumental in these observational tests of infla-tion (Peiris et al. 2003; Barger et al. 2003; Leach & Liddle2003; Kinney et al. 2004). Particularly exciting is WMAP’sdetection of an anti-correlation between CMB temperatureand polarization fluctuations at angular separations θ > 2

(corresponding to the TE anti-correlation seen on scalesℓ ∼ 100 − 150), a distinctive signature of adiabatic fluc-tuations on super-horizon scales at the epoch of decou-pling (Spergel & Zaldarriaga 1997), confirming a fundamen-tal prediction of the inflationary paradigm. This is prob-ably the least ambiguous signature of the acausal physicswhich characterize inflation, unlike the large-angle correla-tions in the CMB temperature which are contaminated by

2 Verde, Peiris & Jimenez

“foregrounds” such as the Integrated Sachs-Wolfe (ISW) ef-fect.

The detection and measurement of a stochasticgravitational-wave background would provide the nextimportant key test of inflation. The CMB polarizationanisotropy has the potential to reveal the signatureof primordial gravity waves (Zaldarriaga & Seljak 1997;Kamionkowski et al. 1997); unlike the CMB temperatureanisotropy, it can discriminate between contributions fromdensity perturbations (a scalar quantity) and gravity waves(a tensor quantity). The polarization anisotropy can be de-composed into two orthogonal modes: E-mode polariza-tion is the curl-free component, generated by both den-sity and gravity-wave perturbations, and B-mode polar-ization is the divergence-free mode, which can only, toleading order, be produced by gravity waves. The primor-dial B-mode signal appears at large angular scales. Onsmaller scales, another effect, which is sub-dominant onlarge scales, kicks in and eventually dominates: weak gravi-tational lensing of the E-mode converts it into B-mode po-larization (Zaldarriaga & Seljak 1998). These considerationsmotivate the current observational and experimental effort,such as on-going ground-based experiments (e.g. QUaD1

(Church et al. 2003), CLOVER2, PolarBeaR3, QUIET4)and planned or proposed space-based and balloon-bornemissions (e.g. Planck5, SPIDER6, EBEX7 (Oxley et al.2005), CMBPol8).

The primordial B-mode anisotropy, if it exists, is atleast an order of magnitude smaller than the E-mode po-larization. This, combined with the difficulty of separatingprimordial B-modes from lensing B-modes as well as po-larized Galactic foregrounds, makes the measurement of apotentially-existing primordial gravity wave background agreat experimental challenge. At the same time, as it is the“smoking gun” of inflation, the primordial B-mode detec-tion and measurement can be regarded as the “holy grail”of CMB measurements.

There are five main obstacles to making a measurementof the primordial B-mode polarization, some being funda-mental, and others being practical complications. The fun-damental complications are: (i) the level of the primordialsignal is not guided at present by theory: there are infla-tionary models consistent with the current CMB data whichpredict significant levels of primordial gravity waves, as wellas models which predict negligible levels; (ii) the signal isnot significantly contaminated by lensing only on the largestscales ℓ <∼ few ×10 where cosmic variance is important; (iii)polarized foreground emission (mostly from our galaxy butalso from extra-galactic sources) on these scales is likely todominate the signal at all frequencies. Practical complica-tions are: (iv) for the signal to be detected in a reasonable

1 QUaD: http://www.stanford.edu/group/quest−telescope/2 CLOVER: http://www.mrao.cam.ac.uk/∼act21/clover.html3 PolarBeaR: http://bolo.berkeley.edu/polarbear/index.html4 QUIET:http://cfcp.uchicago.edu/∼peterh/polarimetry/quiet3.html5 Planck:

http://sci.esa.int/science-e/www/area/index.cfm?fareaid=176 SPIDER: http://www.astro.caltech.edu/∼lgg/spider− front.htm7 EBEX: http://groups.physics.umn.edu/cosmology/ebex/8 CMBPol: http://universe.nasa.gov/program/inflation.html

time-scale, the instrumental noise needs to be reduced wellbelow the photon noise limit for a single detector, so multi-ple detectors need to be used; (v) polarized foregrounds arenot yet well known, and since any detection relies on fore-ground subtraction, foreground uncertainties may seriouslycompromise the goal.

Tucci et al. (2004) consider different foreground sub-traction techniques and compute the minimum detectabler for future CMB experiments for no lensing contamination.Hirata & Seljak (2003) and Seljak & Hirata (2004) showhow the lensing B-mode contamination can be subtractedout in the absence of foregrounds. Calculations of the min-imum detectable primordial gravity wave signature on theCMB polarization data and related constraints on the infla-tionary paradigm have been discussed in the literature [e.g.Knox & Song (2002); Song & Knox (2003); Kinney (2003);Sigurdson & Cooray (2005)].

Bowden et al. (2004) discuss the optimization of aground-based experiment (in particular, focusing on QUaD)and Hu et al. (2003) discuss the instrumental effects thatcan contaminate the detection.

In this work, we take a complementary approach. Wefirst attempt to forecast the performance of realistic CMBexperiments: space-based, balloon-borne and ground based;considering the covariance between cosmological parametersand primordial parameters, effects of foreground contamina-tion, instrumental noise and the effect of partial sky coveragefor ground-based experiments, and then consider what theserealistic forecasts can teach us about inflation.

Given the recent advances in the knowledge of the po-larized foregrounds (Leitch et al. 2005; Benoit et al. 2004)and in detector technology, we then attempt here to an-swer the following questions: How much can we learn aboutthe physics of the early universe (i.e. can we test the infla-tionary paradigm and constrain inflationary models) from arealistic CMB experiment? At what point does instrumen-tal noise and uncertainties in the foregrounds degrade theresults that could be obtained from an ideal experiment (nonoise, no foregrounds)? What will be the limiting factor:foreground subtraction or instrumental noise? How muchcan be learned from a full-sky space-based experiment vs. apartial-sky ground-based one? And finally: Given the largecosts involved in improving the experiment (reducing noiseby increasing the number of detectors and improving fore-ground cleaning by increasing the number of frequency chan-nels), what is the point of diminishing returns?

We start in § 2 by reviewing recent literature on B-modes and inflationary physics, and summarizing present-day constraints on inflationary models. We describe our fore-cast method in § 3 and § 4. In § 5 we report the expectedconstraints on primordial parameters from a suite of exper-iments including instrumental noise and a realistic estimateof foregrounds. These results give us some guidance in opti-mizing future CMB polarization experiments for BB detec-tion and for studying inflationary physics, which we reportin § 6. In § 6 we also draw some conclusions on the con-straints that can be applied to inflationary models based onthe capabilities of these experiments.

CMB polarization and inflation 3

2 B-MODES AND INFLATION

As Peiris et al. (2003) and Kinney et al. (2004) demon-strate, a wide class of phenomenological inflation modelsare compatible with the current CMB data. In the absenceof specific knowledge of the fundamental physics under-lying inflation, an important consideration is therefore tosee what general statements one can make about inflationfrom various types of measurements. It is well-known (e.g.Liddle & Lyth (1993); Copeland et al. (1993b,a); Liddle(1994)) that a measurement of the amplitude of the tensormodes immediately fixes the value of the Hubble parame-ter H during inflation when the relevant scales are leavingthe horizon, or equivalently, the potential of the scalar fielddriving inflation (the inflaton) and its first derivative, V andV ′. The relations between these quantities are given by (seee.g. Lyth (1984))

H ≡ a/a

≃ 1

MPl

√

V

3, (1)

r =2V

3π2M4Pl∆

2R(k0)

(2)

≃ 2V

3π2M4Pl

1

2.95 × 10−9A(k0)

= 8M2Pl

(

V ′

V

)2

, (3)

where MPl ≡ (8πG)−1/2 = mpl/√

8π = 2.4 × 1018 GeVis the reduced Planck energy, r = ∆2

h(k0)/∆2R(k0) and ∆2

h

and ∆2R are the power spectra of the primordial tensor and

curvature perturbations defined at the fiducial wavenumberk0 = 0.002 Mpc−1, where the scalar amplitude A(k0) is alsodefined, following the notational convention of (Peiris et al.2003).

A measurement of the spectral index of the scalar (den-sity) perturbations fixes V ′′, and a potential measurementof the running of the scalar spectral index carries informa-tion about the third derivative of the potential, V ′′′. Further,combining a measurement of the spectral index of tensor per-turbations, nt, with the tensor-to-scalar ratio r allows oneto test the inflationary consistency condition, r = −nt/8,that single-field slow roll models must satisfy. However, aprecise determination of the tensor spectral index is likelyto be even more difficult than a measurement of the tensoramplitude.

The current constraint on the tensor-to-scalar ratio at95% significance from WMAP data alone gives the energyscale of slow-roll inflation as (Peiris et al. 2003):

V 1/46 3.3 × 1016r1/4 GeV; (4)

This is the familiar result that a significant contributionof tensor modes to the CMB requires that inflation takesplaces at a high energy scale. If instead of using CMB onlydata we use the most stringent constraints to date on r, a95% upper limit of r < 0.36, obtained from a compilationof current CMB plus large scale structure data (Seljak et al.2004), one obtains

V 1/46 2.6 × 1016 GeV, (5)

or equivalently, V 1/46 2.2×10−3 mPl. Several authors (e.g.

Lyth (1997); Kinney (2003)) have discussed the physical im-plications of a measurement of primordial B-mode polariza-tion. One can take the point of view that the inflaton φ isa fundamental field and assume the techniques of effectivefield theory, for which the heavy degrees of freedom are in-tegrated out, to be a correct description of the physics ofinflation. In this case, the effective inflaton potential canbe expanded in non-renormalizable operators suppressed bysome higher energy scale, for example the Planck mass:

V (φ) = V0 +1

2m2φ2 + φ4

∞∑

p=0

λp

(

φ

mPl

)p

. (6)

For the series expansion for the effective potential to beconvergent in general and the effective theory to be self-consistent, it is required that φ ≪ mPl. Lyth (1997) showedthat the width of the potential (i.e. the distance traversedby the inflaton during inflation) ∆φ, can also be related tor. The change in the inflaton field during the ∼ 4 e-foldswhen the scales corresponding to 2 6 ℓ 6 100 are leavingthe horizon is

∆φ

mPl∼ 0.39

(

r

0.07

)1/2

. (7)

According to this bound, high values of r require changesin ∆φ to be of the order of mPl; this is in fact only a lowerbound, since at least 50-60 e-folds are required before infla-tion ends (e.g. Liddle & Leach (2003)), meaning that overthe course of inflation, ∆φ could exceed the bound in Eq. 7by an order of magnitude. For the present discussion, the im-portant point is that for a tensor-to-scalar ratio of order 0.1,the field must travel a distance ∆φ ≫ mPl over the courseof inflation, so that the effective field theory description inEq. 6 must break down at some point during inflation. Thisfact has been used to argue, based on self-consistency, that avery small tensor/scalar ratio is expected in a realistic infla-tionary universe, leading to the discouraging conclusion thatthe primordial B-modes in the CMB would be unobservablysmall.

However, arguments based on effective field theory arenot the only way to study the dynamics of inflation. The in-flaton may not be a fundamental scalar field; in fact, as wediscuss later in § 6.2, advances are being made building “nat-ural” particle physics-motivated inflationary models basedon approximate shift symmetries and extra-dimensional se-tups. Also, one can take the approach that, for the predic-tions of single-field inflation to be valid, the only requirementis that the evolution of spacetime is governed by a single or-der parameter. In this approach, one can reformulate theexact dynamical equations for inflation as an infinite hier-archy of flow equations described by the generalized “Hub-ble Slow Roll” (HSR) parameters (Hoffman & Turner 2001;Kinney 2002; Easther & Kinney 2003; Peiris et al. 2003;Kinney et al. 2004). In the Hamilton-Jacobi formulation ofinflationary dynamics, one expresses the Hubble parameterdirectly as a function of the field φ rather than a functionof time, H ≡ H(φ), under the assumption that φ is mono-tonic in time. Then the equations of motion for the field andbackground are given by:

φ = −2M2plH

′(φ), (8)[

H ′(φ)]2 − 3

2M2pl

H2(φ) = − 1

2M4pl

V (φ). (9)

4 Verde, Peiris & Jimenez

Here, prime denotes derivatives with respect to φ. Equation(9), referred to as the Hamilton-Jacobi Equation, allows usto consider inflation in terms of H(φ) rather than V (φ). Theformer, being a geometric quantity, describes inflation morenaturally.

This picture has the major advantage that it allows usto remove the field from the dynamical picture altogether,and study the generic behavior of slow roll inflation withoutmaking assumptions about the underlying particle physics(although the underlying assumption of a single order pa-rameter is still present). In terms of the HSR parametersℓλH , the dynamics of inflation is described by the infinitehierarchy

ℓλH ≡(

mPl

4π

)ℓ (H ′)ℓ−1

Hℓ

d(ℓ+1)H

dφ(ℓ+1). (10)

The flow equations allow us to consider the model spacespanned by inflation using Monte-Carlo techniques. Sincethe dynamics are governed by a set of first-order differentialequations, one can specify the entire cosmological evolutionby choosing values for the slow-roll parameters ℓλH , whichcompletely specifies the inflationary model.

Note, however, two caveats: first, in practice one hasto truncate the infinite hierarchy at some finite order; inthis paper we retain terms up to 10th order. Secondly, thechoice of slow roll parameters for the Monte-Carlo processnecessitates the assumption of some priors for the ranges ofvalues taken by the ℓλH . In the absence of any a priori the-oretical knowledge about these priors, one can assume flatpriors with some ranges dictated by current observationallimits, and the requirement that the potential satisfies theslow-roll conditions. Changing this “initial metric” of slow-roll parameters changes the clustering of phase points on theresulting observational plane of a given Monte-Carlo simu-lation. Thus, the results from these simulations cannot be

interpreted in a statistical way.

Nevertheless, important conclusions can be drawnfrom the results of such simulations (e.g. Kinney (2002);Peiris et al. (2003)); they show that models do not coverthe observable parameter space uniformly, but instead clus-ter around certain attractor regions. And these regions donot correspond especially closely to the expectations fromeffective field theory; they show significant concentrationsof points with significant tensor/scalar ratio.

Recent work by Efstathiou & Mack (2005) shows that,in the inflationary flow picture, there is no well-defined re-lation of the form Eq. 7. However, a reformulation of thebound in Eq. 7 is found if one imposes observational con-straints on the scalar spectral index ns and its running withscale dns/d ln k.

The precise values and error bars on these parametersdepend on the particular data set (or combinations of datasets) used in the parameter estimation, and some data setsyield cleaner and more reliable determinations than others(Seljak et al. 2003). Nevertheless, for our purposes we shalluse the tightest constraints available at present. These comefrom a compilation of CMB data, and large scale struc-ture data from galaxy surveys and Lyman-α forest lines(e.g. Seljak et al. (2004)): at 95% confidence these limits are0.92 < ns < 1.06 and −0.04 < dns/d ln k < 0.03.

Here we illustrate how an empirical relation between ∆φ

and r can be found and show that this relation is somewhatinsensitive to the priors on the ℓλH parameters.

Figures 1 and 2 show the absolute value of ∆φ overthe final 55 e-folds of inflation, plotted against the tensor-to-scalar ratio r. Fig. 1 shows a 2-million point Monte-Carlosimulation of the flow equations, allowing the fourth slow rollparameter, 4λH , to have a wide prior 4λH = [−0.025, 0.025].This prior on 4λH has been (arbitrarily) set to have thesame width as the bound on 3λH allowed by the WMAPdata (Peiris et al. 2003). A wide prior is also allowed on thehigher order HSR parameters. In the simulation shown inFig. 2, the width of the prior on 4λH has been decreasedby a factor of five, 4λH = [−0.005, 0.005]. The higher or-der HSR parameters also have a narrower prior than theprevious case. These choices are motivated by the prejudicethat higher-order parameters in the expansion should havenarrower priors around zero, and by the fact that tighterconstraints on 3λH are possible from CMB and large-scalestructure data compilations or future higher-precision CMBdata.

As was emphasized earlier, the top panels of these fig-ures show that the clustering of points in the two cases isdifferent due to the different initial distributions allowed.The bottom panels show the remaining points after the ob-servational constraints from the above compilation of CMBand large scale structure have been applied. It is seen thatwith this prior in phase-space, there is now a fairly well-defined relation between ∆φ and r, and that even thoughthe distributions in the two simulations with initial priorsof different width are different, this relationship is roughlythe same in both cases. As reported in Efstathiou & Mack(2005), this reformulation of the bound in Eq. 7 is given by

∆φ

mPl≃ 6r1/4. (11)

As Efstathiou & Mack (2005) point out, the relation inEq. 11 is so steep that, to probe models with small field vari-ations where an effective field theory description is likely tobe valid, one needs to be able to detect r 6 10−4. Conversely,a detection of a larger tensor component would support alarge-field model for inflation.

Note that the discussion and the bound stated in Eq. 11apply to non-hybrid models, plotted as dots. Figures 1 and 2also show Monte-Carlo simulations which qualify as hybridinflation models (e.g. Linde (1994)), plotted as stars. Theseindeed show a few small-field models which have significantvalues of r. However, it must be noted that these models in-flate forever till inflation is suddenly ended at some criticaltime by a second field; the observables for the Monte-Carlosimulations for models in this class have been computed ar-bitrarily at 400 e-folds before the end of inflation for thepurposes of these figures - observables computed at a dif-ferent time during inflation would obviously yield differentresults. If a value of r >∼ 10−4 is detected, hybrid modelscan be easily distinguished from large-field models by look-ing at the scalar spectral index: large-field models predict ared spectral index while hybrid models predict a blue index.

We will now try to assess the minimum level of r de-tectable with an idealized and a realistic CMB experiment.

CMB polarization and inflation 5

Figure 1. The absolute value of the range traversed by the scalarfield φ over the final 55 e-folds of inflation, ∆φ, plotted against thetensor-to-scalar ratio r: (top) all models that sustain at least 55e-folds of inflation in a 2-million point Monte-Carlo simulation to10th order of the inflationary flow equations; (bottom) the subsetof models that satisfy current observational constraints on ns anddns/d lnk, as discussed in the text. Hybrid models, which havens > 1, are shown as stars, and the rest as dots (see text). Thissimulation has a wider prior on 4λH and higher order slow rollparameters.

3 METHOD

Here we describe the methods and the notation we will usefor our error forecasts.

3.1 Definitions and Conventions

Following Bond et al. (2000), we define

CXYℓ =

ℓ(ℓ + 1)

2π

∑

m

aX∗ℓmaY

ℓm

2ℓ + 1, (12)

Figure 2. Otherwise identical to Fig. 1, except that the simula-tion shown here has a narrower prior on 4λH and higher orderslow roll parameters.

where aℓm denotes the spherical harmonic coefficients, theXY superscript can be TT, TE,EE or BB, aX

ℓm denotes thespherical harmonic coefficients of the X signal and

Cℓ =ℓ(ℓ + 1)

2πCℓ . (13)

For our forecast calculations, we will assume Gaussianbeams: if ΘF WHM denotes the FWHM of a Gaussian beam,then σb = 0.425 ΘF WHM .

The noise per multipole n0 is given by n0 = σ2pixΩpix

where Ωpix = ΘF WHM × ΘF WHM = 4πfsky/Npix is thepixel (beam) solid angle and Npix is the number of pixels(independent beams) in the survey region and fsky is thefraction of the sky covered by observations. σ2

pix is the vari-ance per pixel, which can be obtained from the detectorsensitivity s as σpix = s/

√Nt. N is the number of detectors

and t the integration time per pixel. With these conventions

the noise bias becomes Nℓ = ℓ(ℓ+1)2π

n0 eℓ2σ2

b .

6 Verde, Peiris & Jimenez

As the receiver temperature Trec is limited to be Trec >hν/k (k is the Boltzmann constant and h is the Planckconstant), the r.m.s temperature ∆Trms is limited to be∆Trms > (TCMB + Trec)/

√∆ν∆t where ∆ν is the band-

width and ∆t the integration time. This limit can be over-come by using multiple detectors; for example QUIET andPolarBeaR will have of the order of 103 detectors.

3.2 Likelihood

Assuming that CMB multipoles are Gaussian-distributed,the likelihood function for an ideal, noiseless, full sky exper-iment can be written as

− 2 lnL =∑

ℓ

(2ℓ + 1)

ln

(

CBBℓ

CBBℓ

)

+ ln

(

CTTℓ CEE

ℓ − (CTEℓ )2

CTTℓ CEE

ℓ − (CTEℓ )2

)

+CTT

ℓ CEEℓ + CTT

ℓ CEEℓ − 2CTE

ℓ CTEℓ

CTTℓ CEE

ℓ − (CTEℓ )2

+CBB

ℓ

CBBℓ

− 3

, (14)

where CXYℓ is the estimator for the measured angular power

spectrum. The likelihood (14) has been normalized with re-spect to the maximum likelihood, where CXY

ℓ = CXYℓ .

In the case of a more realistic experiment (with partialsky coverage, and noisy data) we obtain:

− 2 lnL =∑

ℓ

(2ℓ + 1)

fBBsky ln

(

CBBℓ

CBBℓ

)

+√

fTTskyfEE

sky ln

(

CTTℓ CEE

ℓ − (CTEℓ )2

CTTℓ CEE

ℓ − (CTEℓ )2

)

+√

fTTskyfEE

sky

CTTℓ CEE

ℓ + CTTℓ CEE

ℓ − 2CTEℓ CTE

ℓ

CTTℓ CEE

ℓ − (CTEℓ )2

+ fBBsky

CBBℓ

CBBℓ

− 2√

fTTskyfEE

sky − fBBsky

(15)

In Eq. 15, CXYℓ = CXY

ℓ +NXYℓ where Nℓ denotes the noise

bias, CXY denotes the measured angular power spectrumwhich includes a noise bias contribution, fXY

sky denotes thefraction of sky observed, X, Y = T, E,B and we haveallowed different fraction of the sky to be observed in T , Eand B.

Note that this equation accounts for the effective num-ber of modes allowed in a partial sky, but does not accountfor the mode correlation introduced by the sky cut whichmay smooth power spectrum features. The details of themode-correlation depend on the specific details of the mask,but for ℓ greater than the characteristic size of the surveyour approximation should hold.

The expression for the likelihood (15) is only valid for asingle frequency experiment. The exact expression for multi-frequency experiments which takes into account for bothauto- and cross- channels power spectra cannot be writtenin a straightforward manner in this form. However, for Nchan

frequencies with the same noise level, considering both autoand cross power spectra is equivalent to having one effec-tive frequency with an effective noise power spectrum lower

by a factor Nchan. This can be understood in two ways. Ifone were to combine Nchan independent maps, the resultingmap will have a noise level lower by a factor

√Nchan; thus

the noise power spectrum will be that of an auto-power spec-trum, reduced by a factor Nchan. One could instead computeauto and cross Cℓ and then combine them in an minimumvariance fashion. Let us assume Gaussianity for simplicity.When combining the Cℓ, the covariance matrix in the ab-sence of cosmic variance will be:

Σii′jj′ =1

2l + 1(ninjδij×ni′nj′δi′j′+ninj′δij′×ni′njδi′j)(16)

where i, j run through the channels, ninj = N ij and δij

denotes the Kronecker delta. Note that this has been com-puted for the no cosmic variance case, as there is no cosmicvariance involved when combining Cℓ’s computed from thesame sky at different wavelengths). Thus, if Neff is the samefor all channels then the covariance between, say, Cℓ, V V &Cℓ, V V is twice that between say Cℓ, V W & Cℓ, V W ; the co-variance between Cℓ, V V & Cℓ, WW and Cℓ, V W & Cℓ, QV

etc. is zero. The minimum variance combination will thusyield an effective noise power spectrum of N/Nchan.

We can generalize the above considerations to the casewhere the different channels have different noise levels. Thisis useful for example in the case where each channel has adifferent residual foreground level, which, as we will showbelow, can be approximated as a white noise contributionin addition to the instrumental noise. The optimal channelcombination would be:

Cl =

∑

i,j,j>iwi,jC

i,jl

∑

i,jwi,j

(17)

where wi,j are weights and are given by [NiNj1/2(1+δij)]−1.

The resulting noise will be given by

Neff =

(

∑

i,j,j>i

1

Neff,iNeff,j12(1 + δij)

)−1/2

(18)

3.3 Error calculation

The Fisher information matrix is defined as (Fisher 1935):

Fij =

⟨

− ∂2L

∂αiαj

⟩

|α=α (19)

where L ≡ lnL and αi, αj denote model parameters. TheCramer-Rao inequality says that the minimum standard de-viation, if all parameters are estimated from the same data,is σαi

> (F−1)1/2ii and that in the limit of a large data set,

this inequality becomes an equality for the maximum likeli-hood estimator.

For our error forecast we use the Fisher matrix ap-proach. Thus here, α denotes a vector of cosmologi-cal parameters (we consider r, n, nt, dn/d ln k, Z, Ωbh

2 ≡wb, Ωch

2 ≡ wc, h, Ωk, A(k0), where Z ≡ exp(−2τ ) ), α de-notes the fiducial set of cosmological parameters, αi beingthe ith component of that vector, and L is given by Eq. 15.For the fiducial parameters, we will consider three cases forr, r = 0.01, 0.03, 0.1 and r = 0.001, r = 0.0001 as a specialcases; for the other cosmological parameters we use n = 1,nt = −r/8, d lnn/d ln k = 0, exp(−2τ ) = 0.72 (correspond-ing to τ = 0.164), Ωbh

2 = 0.024, Ωch2 = 0.12, h = 0.72,

CMB polarization and inflation 7

Ωk = 0, A(k0) = 0.9, in agreement with WMAP first yearresults (Spergel et al. 2003).

Following Spergel et al. (2003), the pivot point for nand A is at k = 0.05 Mpc−1 and the tensor to scalar ratior is also defined at k = 0.05 Mpc−1. Making use of the con-sistency condition and assuming no running of the scalar ortensor spectral indices, r(k0) can be related to the r definedat any other scale k1 by

r(k1) = r(k0)(

k1

k0

)−r(k0)/8+1−ns(k0)

. (20)

Therefore, for our choice of a scale-invariant scalar powerspectrum, e.g. r(0.05) = 0.1 corresponds to r(0.002) = 0.104and our results are thus not significantly dependent on thechoice of the pivot point.

Note that for a fixed value of r, the height of the BB“reionization bump” depends on τ approximately as τ 2,but the position of the “bump” maximum also varies. Theτ determination from WMAP has a 1-σ error of ∼ 0.07(Spergel et al. 2003). Thus for a fiducial value of τ ∼ 0.1,one sigma below the best fit, the detectability of r = 0.03(r = 0.1) from ℓ <∼ 20 is roughly9 equivalent to the de-tectability of r = 0.01 (r = 0.03) in the best fit model. Onthe other hand, for a fiducial value of τ one sigma above thebest fit, the detectability of r = 0.01 (r = 0.03) from ℓ <∼ 20is equivalent to the detectability of r = 0.03 (r = 0.1) inthe best fit model. This consideration affects only large skycoverage experiments which probe the reionization bump inthe BB spectrum at ℓ < 10, and does not affect the smallerscale ground based experiments. We therefore only calculatethe case of the one-sigma lower bound fiducial τ = 0.1 forthe space-based test case with r = 0.001 and the balloon-borne large angle experiment SPIDER. As a higher value forτ boosts the BB signal, we investigate this lower value of τin order to be conservative in our predictions.

We use all the power spectrum combinations as specifiedin Eqs. 14 and 15, computed to ℓ = 1500 and using theexperimental characteristics specific to each experiment westudy. For the partial sky experiments we add WMAP priorson n, dn/d ln k, Z, Ωbh

2, Ωch2, h, and A.

We use the “three point rule” to compute the secondderivative of L when i = j:

∂2L

∂2αi=

L(α−δαi) − 2L(α) + L(α+δαi

)

δα2i

. (21)

Here, by L(α) we mean L evaluated for the fiducial set ofcosmological parameters. α+δαi

denotes the vector of cos-mological parameters made by the fiducial values for all el-ements except the ith element, which is αi + δαi; δαi is asmall change in αi.

For i 6= j we use:

∂2L

∂αi∂αj=

[L(α+δαi+δαj) − L(α+δαi−δαj

)]

[L(α−δαi+δαj) − L(α−δαi−δαj

)]

/(2δαi2δαj) (22)

where α+δαi−δαjdenotes a vector of cosmological pa-

rameters where all components are equal to the fiducial cos-

9 Note that since the “bump” location changes as τ changes thisargument is only qualitative.

mological parameters, except components i and j which areαi + δαi and αj − δαj respectively.

For high multipoles the Central Limit Theorem ensuresthat the likelihood is well approximated by a Gaussian.Thus, if one also neglects the coupling between TT , TEand EE, the high ℓ Fisher matrix can be well approximatedby

− 2〈 ∂ lnL∂αi∂αj

〉|α=α =∑

P,ℓ

∂CPℓ

∂αiΣ−1

ℓℓ′∂CP

ℓ′

∂αj, (23)

where P = TT, EE, BB,TE, and Σ denotes the covariancematrix. Therefore the Fisher matrix calculation can be sped

up greatly by precomputing∂CP

ℓ

∂αi. However, since most of the

r signal comes from low ℓ, we will not use this approximationhere.

3.4 Detection vs Measurement

There are two different approaches in reporting an experi-ment’s capability to constrain r. In the first approach, oneconsiders the null hypothesis of zero signal (i.e. r = 0); thenone asks with what significance a non-zero value of r couldbe distinguished from the null hypothesis. This approachgives the statistical significance of a detection, but it doesnot yield a measurement of r. To obtain a measurement ofr with an error-bar, a different approach is needed. In thiscase, the value of r is obtained via a Bayesian maximumlikelihood analysis, and the cosmic variance contribution fornon-zero r is included in the error calculation. Since a mea-surement of r is needed to constrain inflationary models,here we follow the second approach. Thus we do not reportthe minimum value of r that can be distinguished from zero,but error-bars for several fiducial r values.

4 FOREGROUNDS

As anticipated in the introduction, foregrounds are one ofthe main obstacles to the detection and measurement of pri-mordial B-modes: they are likely to dominate the signal atall frequencies. And, while foreground intensities are rela-tively well known, polarized foregrounds are not. Here wewill neglect the effect of foregrounds on the CMB tempera-ture, as foreground contamination in the temperature dataat the angular scales considered is much smaller than thecontamination to the polarized signal and foreground clean-ing is expected to leaves a negligible contribution in thetemperature signal (Bennett et al. 2003b).

CMB polarized foregrounds arise due to free-free, syn-chrotron, and dust emission, as well as due to extra-galacticsources such as radio sources and dusty galaxies. Here, wewill consider only synchrotron and dust emission; in fact,free-free emission is expected to be negligibly polarized, andthe radio source contribution remaining after masking thesky for all sources brighter than ∼1 Jy is always well belowother foregrounds at large angular scales. E.g. Tucci et al.(2004)) find that this residual point source contribution is<∼ 1% of the other foregrounds in the power for ℓ <∼ 200.Note, however, that at ℓ >∼ 100, this contribution that weneglect can be as important in the BB power spectrum as

8 Verde, Peiris & Jimenez

the lensing signal (see Tucci et al. (2004)), and therefore canseriously hinder the delensing implementation.

We also neglect the contribution from dusty galaxies.

4.1 Synchrotron

Free electrons spiral around the Galactic magnetic field linesand emit synchrotron radiation. This emission can be upto 75% polarized, and is the main CMB foreground at lowfrequencies. We model the power spectrum of synchrotron as(e.g. Tegmark et al. (2000); de Oliveira-Costa et al. (2003)):

CS,XYℓ = AS

(

ν

ν0

)2αS(

ℓ

ℓ0

)βS

(24)

where we have assumed that Galactic synchrotron emis-sion has the same amplitude for E and B. The unpolar-ized synchrotron intensity has αS ∼ −3 with variationsacross the sky of about ∆αS ∼ 0.15 (e.g. Platania et al.(1998, 2003); Bennett et al. (2003b)) and βS ≃ −3 (e.g.Giardino et al. (2002) and references therein) with variationacross the sky 10. Information on polarized synchrotron islimited at present to frequencies much lower than CMB fre-quencies (e.g. Carretti et al. (2005a); Carretti et al. (2005a)and mostly low Galactic latitudes).

Recent estimates for βS for polarized synchrotron emis-sion (Tucci et al. 2002; Bruscoli et al. 2002) show spatialand frequency variations. For example, Bruscoli et al. (2002)study the angular power spectrum of polarized Galactic syn-chrotron at several frequencies below 3 GHz and at severalGalactic latitudes, and find a wide variation mostly due toFaraday rotation and variations in the local properties of theforegrounds. Bernardi et al. (2004) find that at high galacticlatitudes there is no significant dependence of the slope onfrequency. Here we consider βS = −1.8 for both E and B,consistent with e.g. Baccigalupi et al. (2001); Bruscoli et al.(2002); Bernardi et al. (2004, 2003).

We take the reference frequency of ν0 = 30 GHz andthe reference multipole ℓ0 = 300 so that we can use theDASI 95% upper limit (Leitch et al. 2005) of 0.91 µK2 toset the amplitude AS. Since the normalization is set at ℓ =300 and most of the r signal is at ℓ < 100, a choice of aflatter βS would have the effect of reducing the synchrotroncontamination at low ℓ.

We will consider three cases: an amplitude equal to theDASI limit (pessimistic case) and half of that amplitude(reasonable case). Recent results by Carretti et al. (2005a)show a detection of synchrotron polarized emission in a cleanarea of the sky at 2.3 GHz. The amplitude of this signal isabout an order of magnitude below the DASI upper limit.We will consider this amplitude (10% of the DASI limit)as a minimum amplitude (optimistic case) achievable onlyfor partial sky experiments which can look at a clean patchof the sky. Unless otherwise stated we will assume the pes-simistic case.

10 At low Galactic latitudes, the synchrotron index shows a flat-tening, but we assume here that the mask will exclude those re-gions

4.2 Dust

We ignore a possible anomalous dust contribution (e.g.Finkbeiner et al. (2004); Draine & Lazarian (1998)) as itis important only at frequencies below 35 GHz or so(Lazarian & Finkbeiner 2003). We model the dust signal as:

CD,XYℓ = p2AD

(

ν

ν0

)2αD(

ℓ

ℓ0

)βXYD

[

ehν0/KT − 1

ehν/KT − 1

]2

(25)

assuming the temperature of the dust grains to be constantacross the sky, T = 18K. We take αD = 2.2 (Bennett et al.2003b), and set our reference frequency to be ν0 = 94 GHz.The amplitude AD is given by the intensity normalization ofFinkbeiner et al. (1999), extrapolated to 94 GHz, and p isthe polarization fraction. Since we want to use the Archeopsupper limit for the diffuse dust component (Benoit et al.2004) of p = 5% at ℓ = 900 (roughly corresponding tothe resolution in Benoit et al. (2004)), we set ℓ0 = 900.Archeops finds αD = 1.7; with our choice the extrapola-tion of the dust contribution at higher frequency is slightlymore conservative. Since a weak Galactic magnetic field of3 µG already gives a 1% (Padoan et al. 2001) polarization,the polarization fraction should be bound to be between1–5%. Unless otherwise stated, we will assume a 5% po-larization fraction. For CXY,D

ℓ we follow Lazarian & Prunet(2002); Prunet et al. (1998): βEE

D = −1.3, βBBD = −1.4,

βTED = −1.95, βTT

D = −2.6 (in agreement with the measure-ment of starlight polarization of Fosalba et al. (2002)).

As parameters like βXYD , βXY

S , αD, αS and p may showspatial variations, the optimal frequency band may be dif-ferent for full sky and partial sky experiments and may varybetween different patches of the sky.

4.3 Propagation of foreground subtraction errors

The foreground treatment presented above is quite simplis-tic and cannot capture the foreground properties completely.However, we will not be using these estimates to subtractthe foreground from the signal. Instead, we assume that fore-ground subtraction can be done correctly down to a givenlevel (e.g. 1%, 10%), and use these foreground models topropagate the effects of foreground subtraction residuals intothe resulting error-bars for the cosmological parameters. Inother words the modeling enters in the error-bars, not inthe signal. Therefore, as long as the foreground assumptionsare reasonable, and foregrounds can be subtracted at the as-sumed level, the results are relatively insensitive to the de-tails of the foreground model. Of course, one cannot know inadvance whether the foreground will be subtracted at thatlevel. For this reason we considered several different options(10%, 1% etc..).

To propagate foreground subtraction errors we pro-ceed as follows. We assume that foregrounds are sub-tracted from the maps via foreground templates or usingmulti-frequency information with techniques such as MEMs(e.g. Bennett et al. (2003b)), ICA (Baccigalupi et al. 2004;Stivoli et al. 2005). We write the effect on the power spec-trum of the residual Galactic contamination as an additional“noise-like” component Cres,fg,XY

ℓ composed by a term pro-portional to the foreground Cℓ and a noise term:

Cres,fg,XYℓ (ν) = Cfg,XY

ℓ (ν)σfg,XY

CMB polarization and inflation 9

+ Nfg,XY

(

ν

νtmpl

)−2〈αfg〉

(26)

where fg denotes dust (D) or synchrotron (S), 〈αfg〉 de-notes the average value of the spectral index, and σfg,XY

quantifies how good the foreground subtraction is (e.g.σfg,XY =0.01 for a 1% residual in the Cℓ). Cfg,XY

ℓ is givenby Eq. 24 for synchrotron and Eq. 25 for dust; Nfg,XY de-notes the noise power spectrum of the template map. As anestimate, we assume that the noise spectrum in the tem-plate maps is a white noise equal to the noise spectrum ofone of the channels reduced by a factor 1/2×n(n−1)/2. Thefactor of 1/2 arises from the fact that foreground templatesare effectively obtained by subtracting maps at two differentfrequencies (which increases the noise in the map by a factor∼

√2), and that there are n(n − 1)/2 map pairs. Although

the map pairs are not all independent, we expect that ex-periments will have some channels both at higher and lowerfrequencies than those devoted to cosmology, to be used forforeground studies. We thus consider this a reasonable esti-mate for the template noise.

νtmpl denotes the frequency at which the template iscreated. For this forecast we assume that νtmpl coincideswith the frequency of the experiment channel where the fore-ground contamination is highest. To justify the ansatz Eq.(26), consider the result from Tucci et al. (2004) (their §3and Appendix).

Cres,fg,XYℓ =

(ln(ν0/ν))2

16π

×∑

ℓ1

(2ℓ1 + 1)Cfg,XYℓ1

∑

ℓ2

(2ℓ2 + 1)Cαfg

ℓ2

(

ℓ ℓ1 ℓ22−2 0

)2

+ CXY,∆fgℓ

(

ν

ν0

)−2〈αfg〉

(27)

where Cαfg

ℓ denotes the power spectrum of the spectral in-dex error. If the power spectrum of the spectral index er-ror has the same ℓ dependence as the total intensity power(that is ∼ ℓ−3 for synchrotron and ∼ ℓ−2.6 for dust), neglect-ing the logarithmic dependence on ν we obtain Eq. 26. Weassume that the foreground template is computed for the“cosmological” frequency where the foreground contamina-tion is highest. For example, for 5 different frequency bandsat (30, 50, 90, 100, 200 GHz), the synchrotron template willbe computed for the 30 GHz band and the dust templatefor the 200 GHz band.

4.4 Delensing

Several different techniques to reconstruct the gravitationallensing potential from CMB data and reduce or remove lens-ing contamination to the BB signal have been proposed(Hu & Okamoto 2002; Kesden et al. 2002; Knox & Song2002; Okamoto & Hu 2003; Kesden et al. 2003). However,in presence of instrumental noise the contamination cannotbe easily removed. In addition foreground emission is ex-pected to be highly non-Gaussian and the performance ofdelensing techniques have not been explored in the presenceof non-Gaussian foregrounds. In fact, the separation of theprimordial gravitational wave contribution is achieved ex-ploiting the non-Gaussianity of the lensed CMB. Thus, thenon-Gaussian nature of the foregrounds will degrade this re-

construction and limit the delensing performance. To includethe above considerations, forecast whether delensing cansuccessfully be implemented and can improve constraintson r we proceed as follows. If foregrounds are neglected, weassume that the delensing technique can be applied only inthe signal-dominated regime, and that in this regime theprocedure can reduce the BB-lensing signal only down tothe level of the instrumental noise power spectrum. This as-sumption is well motivated by results of e.g., Seljak & Hirata(2004). When including foregrounds, we assume that delens-ing can be implemented only if the BB power spectrum ofthe combination of foreground emission and foreground tem-plate noise is below 10% of the BB lensing power spectrum,and that the BB-lensing signal can be reduced only down to10% of the foreground signal (emission plus template noise).We deem this to be a reasonable assumption: a percent levelof foreground contamination in the angular power spectrumis probably close to the minimum achievable and it corre-sponds to a 10% level in the maps. The residual foregroundwill be highly non-Gaussian thus limiting the delensing tothis level . When considering a realistic case with noise andforegrounds, for every ℓ we consider whichever effect is mostimportant.

5 CASES WE CONSIDER

We consider several different experimental settings. Firstan ideal experiment (IDEAL), then a more realistic full-skysatellite experiment (SATEL) and finally, ground-based andballoon-born partial sky experiments. For all these cases, wecompute the marginalized errors on the parameters using theFisher matrix approach illustrated above. In the main textof the paper we report errors for the parameters relevantfor inflation and parameters degenerate with these (i.e. r,n, nt, dn/d ln k and Z). The errors on the other cosmologi-cal parameters are reported in the Appendix. For each casewe start by considering ten cosmological parameters (r, n,nt, dn/d ln k, Z, wb wc, h, Ωk and A). We do not considercalibration uncertainties, even though they propagate intothe error on the amplitude. As exploring non-flat models iscomputationally expensive when we find that ΩK is only de-generate with h, we impose flatness (FLT). This constraintdoes not affect the estimated error on the remaining cosmo-logical parameters. Finally, we report errors obtained withand without imposing the consistency relation (nt = −r/8,CR).

5.1 Ideal experiment

Let us consider an ideal experiment, covering the full sky,with no instrumental noise and no foregrounds. We considerℓmax=1500 because smaller angular scales will be observedfrom the ground and, most importantly, because higher mul-tipoles in the temperature are affected by secondary effectswhile in the polarization ℓ > 1500 does not add significantcosmological information.

Since Hirata & Seljak (2003); Seljak & Hirata (2004)showed that the lensing contribution to the BB spectracan be greatly reduced and, in the absence of instrumentalnoise and foregrounds, completely eliminated, we considertwo cases: the BB spectrum with lensing (L) and without

10 Verde, Peiris & Jimenez

Figure 3. EE and BB signal for r = 0.03 fiducial model, maximum estimated foreground level and foreground residuals for 1%foreground subtraction at 90 GHz. Synchrotron is normalized to the DASI upper limit at ℓ = 300 30GHz assuming a frequency dependenceof ν−3. Dust is normalized so that at 94 GHz, ℓ = 900 dust is 5% polarized. The template noise is that expected for the hypotheticalsatellite experiment.

Table 1. 1-σ errors for an ideal experiment, including lensing(L), with no lensing (NL).

r ∆r ∆n ∆nt ∆dn/d ln k ∆Z

0.01 0.001 0.0017 0.056 0.003 0.003L 0.03 0.0027 0.0017 0.047 0.0036 0.003

0.1 0.006 0.002 0.035 0.0035 0.0035

0.01 0.000021 0.0021 0.0019 0.0038 0.0038NL 0.03 0.000063 0.0021 0.0019 0.0038 0.0038

Table 2. Ideal experiment, ℓmax = 1500, nt fixed by consistencyrelation (CR)

r ∆r L ∆r NL

0.01 4.5 × 10−4 1.6 × 10−5

0.03 7.4 × 10−4 4.9 × 10−5

(NL) (i.e. with the lensing contribution perfectly subtractedout).

We report the results in table Table 1 In the L case, dueto the lensing contamination, the same error can be achievedby considering only ℓ 6 300 for BB. Errors on other relevantcosmological parameters are reported in Table 1. We findthat r is mainly degenerate with nt; by fixing nt, the erroron r is greatly reduced (see Table 2), but the errors on theother cosmological parameters do not change significantly.

If lensing can be completely subtracted from the BBspectrum, we obtain that the error bars on r improve bymore than an order of magnitude (see Table 6).

5.2 Space-based experiment

We next consider a full-sky (space-based) experiment witha beam of FWHM 8 arcminutes, 5 frequency channels(Nchan = 5) at 30, 50, 70, 100, 200 GHz, and a sky cov-

Table 3. Realistic satellite experiment, no foreground, no lensingsubtraction, marginalized errors, ℓmax = 1500. The error on therunning is ∆dn/d lnk = 0.0046 in all these cases.

case r ∆r ∆n ∆nt

0.01 0.003 0.0023 0.098NO FG 0.03 0.0048 0.0023 0.069

0.1 0.01 0.0023 0.056

0.01 0.0011 0.0023 –NO FG 0.03 0.0017 0.0023 –

CR 0.1 0.0028 0.0023 –

erage of 80%. Since the Galactic emission is highly polar-ized we consider that a realistic Galactic cut will excludeabout 20% of the sky. We start by assuming a noise levelof 2.2 µK per beam per frequency channel. This noise levelis achievable with the next generation space-based experi-ments, and is well-matched to measure the first peak in apossible r = 0.01 gravity wave signal in the cosmic variance-dominated regime. We will then explore the effect of a higher(lower) noise level on the errors on r. As before, our maxi-mum multipole will be ℓmax = 1500. Finally we will explorehow the constraints can be improved by implementing de-lensing.

We first consider an idealized case where there are noforegrounds. The constraints on r, n, and nt are reportedin table 3, the error on dn/d ln k is ∆dn/d ln k = 0.0046 inall these cases. When imposing flatness these constraints arevirtually unchanged. Marginalized constraints on the otherparameters are reported in the Table 6.

Thus, we see that in the absence of foreground con-tamination, a realistic experiment with this noise level canachieve constraints close to those for an ideal experiment forour fiducial values of r. We find that the noise level can behigher by a factor 10 before the constraints get significantlydegraded. In particular we find that for r = 0.03 a noise level

CMB polarization and inflation 11

of 22 µ K per beam increases the error on r by an order ofmagnitude with respect to the zero-noise case.

Since the noise is not the limiting factor for the valuesof r considered here, we conclude that reducing the sky cov-erage of such an experiment to reduce the noise level wouldnot improve the signal to noise.

5.2.1 Delensing

To take into account of the possibility of improving thesignal-to-noise by applying the “delensing” (DL) techniquewe proceed as follows. We start by ignoring foregrounds andconsider only the effect of instrumental noise. Since in thenoise-dominated regime (i.e. when the power spectrum of thenoise is greater than the BB-lensing power spectrum), weassume that delensing cannot be successfully implemented,we find11 that for our realistic satellite experiment, delens-ing does not significantly improve the constraints on therelevant parameters. In other words, for the noise level con-sidered, the signal-dominated regime for lensing is too smallto help improve constraints significantly.

Thus, in the absence of foregrounds, reducing the skycoverage to lower the noise would help such a space-basedexperiment to access the primordial signal at the “peak”.However, partial sky experiments can be done much morecheaply from the ground or from a balloon and can reacheven lower noise levels. We shall see below how these con-clusions can change in the presence of foregrounds.

5.2.2 Foregrounds

We find that even in the pessimistic case for the foregroundamplitude (for both dust and synchrotron), if foregroundscan be subtracted at the 1% level (σfg,XY = 0.01) and chan-nels can be optimally combined to minimize the effect of theresidual contamination, the additional noise-like componentCres

ℓ is comparable to or smaller than the instrumental noiseof 2.2 µK per beam. In particular, we find that the totaleffective noise (that is, instrumental noise plus foregroundsubtraction residuals) is higher than the instrumental noisecontribution by a factor ∼ 3 at ℓ = 2, but only a factorof ∼ 1.4 at ℓ = 10 and ∼ 1.1 at ℓ > 100. As a result, theconstraints on cosmological parameters are not significantlyaffected by the presence of a residual foreground contami-nation at this level. Note that we do not propagate into theerror bars a possible uncertainty on the residual foregroundcontamination, as this will be a higher-order effect in ourapproach; in other words we assume that the uncertaintyin the residual foreground contamination is negligible com-pared with the residual contamination level. When dealingwith realistic data the validity of this assumption will needto be assessed.

In order for the foregrounds to be subtracted at thepercent level, a drastically improved knowledge of the am-plitude and the spectral and spatial dependence of polarized

11 For greater computational speed we have fixed Ωk to be zerofor this calculation. We find that this affects only the error onthe Hubble parameter which is not the main focus of the presentwork.

foreground emission is needed. This can be achieved by ob-serving the polarized foreground emission on large areas ofthe sky, at high galactic latitudes, with high resolution andhigh sensitivity and at multiple wavelengths, both at higherand lower frequencies than the “CMB window”. Most likely,a combination of approaches will be needed. For example,the Planck satellite will observe the full sky polarization upto 350 GHz and down to 30 GHz. The proposed BLAST-Pol (M. Devlin, private communication) will survey smallerregions of the sky at higher sensitivity and at higher frequen-cies, enabling one to extrapolate the dust emission more se-curely. The ability to create accurate foreground templatesand to reduce foreground contamination will depend cru-cially on the success of these efforts.

As a foreground residual of 1% may be too optimistic,we also consider cases with a 10% foreground residual in thepower spectrum. We find that, while for level of foregroundresiduals of 10constraints on the recovered parameters de-grade rapidly.

Our fiducial values for r (0.01, 0.03 and 0.1) have beenchosen because they are accessible to experiments in the verynear future. A detection of such a large primordial tensorcomponent would support a large-field model (see also thediscussion of hybrid inflation models in §2). Following theconsiderations of Efstathiou & Mack (2005), it would implythat, if the inflaton is a fundamental field, the effective fieldtheory description is not valid, i.e. that Eq. 6 is not a self-consistent description of inflation.

Thus, we also investigate lower values of r of 10−3 and10−4 which are closer to satisfying the effective field theorydescription (see Table 4). We find that a three-σ detection ofr = 10−3 is realistically achievable if the foreground contam-ination is lower than the DASI upper limit and if foregroundcleaning can reduce contamination at least to the 1% levelin the power spectrum. However a value of r = 10−4 can-not be detected by this realistic experiment given our noiselevel and estimates of foregrounds. To assess whether thelimiting factor for detecting such a gravity wave signal isthe noise level or foreground contamination, we computedthe expected errors for a case with no noise, synchrotronamplitude 1/2 of the DASI limit, dust polarization fractionof 5% and foreground subtraction at the 1% level. Imposingthe consistency relation we find that ∆r = 6 × 10−5, noteven a two-σ detection. This implies that the main obstacleto detecting such a small value of r will come from Galacticforegrounds. To assess quantitatively how much the r detec-tion could be degraded by a lower value of τ we also reportthe errors for a fiducial model with r = 0.01 and τ = 0.1.

As the delensing implementation in this case is limitedby foregrounds residuals, not by the noise level, it would nothelp for a space-based experiment to reduce the sky coverageby a factor of a few, say, to reduce the noise levels. As wewill see below, delensing could be implemented by targetinga small region of the sky with particularly clean foregrounds.However, this can be done more easily and cheaply from theground.

Finally we note that, although we used the full combi-nation of T , E and B data, the signal for r comes from theBB signal. In fact, we find that if we consider our realisticsatellite case but only T and E-mode polarization data, onlyupper limits can be imposed on r for r < 0.1.

12 Verde, Peiris & Jimenez

Table 4. Realistic satellite experiment including foregrounds forr 6 10−3 and flat cosmology. We have assumed that synchrotronemission amplitude is 50% of DASI limit, a dust polarization frac-tion 5% and that foregrounds can be subtracted to 1% in thepower spectrum. For comparison we also report a case for no in-strumental noise. In the latter case, the error on r is reducedonly by a factor ∼ 2, indicating that foregrounds contaminationbecomes the limiting factor.

case r ∆r ∆n ∆nt ∆dn/d ln k

SAT 0.0001 0.00019 0.0023 0.43 0.00460.001 0.0013 0.0023 0.31 0.0046

+CR 0.0001 0.00012 0.0023 – 0.00460.001 0.0003 0.0023 − 0.0046

SAT τ = 0.1 0.001 0.0020 0.002 0.47 0.0045+CR 0.001 0.00043 0.002 – 0.0045

NO NOISE 0.0001 0.00007 0.0019 0.17 0.0037+CR 0.0001 0.000057 0.0019 – 0.0037

5.3 Ground-based and balloon-borne experiments

While a space-based experiment may be a decade away, inthe shorter term, partial-sky ground-based experiments willbe operational in the next few years. Ground-based exper-iments are less expensive and can achieve a lower level ofnoise contamination at the expense of covering a smaller re-gion of the sky. However, as the primordial signal is minimalat 10 < ℓ < 30, the signal-to-noise is expected to scale onlyas the square root of the sky fraction for experiments cover-ing more that about 200 square degrees (for the same noiselevel, frequency coverage and angular resolution).

For r = 0.03, the primordial B-mode signal dominatesthe lensing one on scales of 5-10 degrees (ℓ <∼ 70), but forr = 0.01, it dominates only on scales larger than few × 10degrees (ℓ <∼ 10). So ground-based experiments may needto target a clean patch of the sky and will need to rely ondelensing to detect r values <∼ 0.03.

On the other hand, the amplitude of the large-scale B-mode signal strongly depends on the value for τ ; thus alimit on r from these scales is somewhat more cosmology-dependent.

Here we consider several ground-based and balloon-borne experiments; the full set of experimental specificationsassumed for each of these are reported in the Appendix (Ta-ble 7).

Continuously changing the sky coverage, frequency cov-erage and number of frequencies, number of detectors andangular resolution for these experiments will be prohibitive.But we want to concentrate on experimental setups that canbe made with next generation technology; hence we reporthere results for a few selected experimental designs that wefound already make the best use of next generation tech-nology. We will then consider small variations around thesesetups. We will also discuss in the text results for selectedcombinations of r and foreground assumptions and reportall other cases considered in Table 6.

Firstly, we consider two instruments with the specifi-cations of the ground-based experiments QUIET and Po-larBeaR (based on HEMPT and bolometer technology re-spectively), and two possible combinations of these ex-periments, ensuring wider frequency coverage. Of these,

QUIET+PolarBeaR is a straight combination of the fre-quency channels and noise properties of QUIET and Polar-BeaR. QUIETBeaR is a hypothetical experiment with thedetectors of QUIET with added channels at the PolarBeaRfrequencies, but with the QUIET 90 GHz noise properties,designed to observe a significantly smaller area of the sky.The motivation behind this combination is to check the im-provement in constraints in an experiment where it is possi-ble to successfully implement delensing. Finally, we considera balloon-based experiment with the properties of SPIDER(J. Ruhl private comm.).

The ground-based experiments considered above obtaintheir r constraint principally by going after the primordialBB “first peak” in the ℓ–range 70-100. On the other hand,the larger sky coverage of the balloon-borne experimentmakes it possible for it to access the “reionization bump”at large scales, where the primordial signal may dominatesignificantly over the weak lensing, depending on the valueof τ . Therefore, this selection of experiments pretty muchcover the possibilities of what can be done without going tospace.

We find that QUIET can detect r = 0.1 at the 3-σ levelwhen imposing the consistency relation, if foregrounds canbe subtracted at the 1% level (see Table 5). Alternatively ifforegrounds can only be cleaned at the 10%, the experimentshould observe a clean patch of the sky (with minimal fore-ground emission) needs to be observed to achieve the samesignal-to-noise.

Since we assume that delensing can be implementedonly where the B-lensing power spectrum is 10 times higherthan the foreground emission and template noise, we findthat delensing cannot be implemented for our pessimisticforeground levels. It could be implemented for a clean patchof the sky where the synchrotron amplitude is 1/10 of theDASI limit and dust polarization fraction is ∼ 1%. In thiscase, the limiting factor for delensing becomes the noise inthe foreground templates. We find that the error-bars arenot significantly changed and r = 0.01 is still out of reach(Table 6).

An experiment like PolarBeaR can improve on theseconstraints on r if the dust contamination is minimal (1% ofdust contamination according to our assumptions; Table 5).

However, an aggressive level of foreground cleaning,as the one considered here, can realistically be achievedonly with a wider frequency coverage well below andabove the cosmological window. Moreover, additional fre-quencies enable one to reduce the foreground templatenoise and make delensing possible. The QUIETBeaR andQUIET+PolarBeaR channel combination, with the additionof higher frequencies, improves dust cleaning; lower frequen-cies will be useful for improving synchrotron subtraction. Wefind that the QUIET+PolarBeaR combination in a cleanpatch of sky (50% of DASI limit and 1% dust polarization)with good foreground subtraction can detect r = 0.1 at bet-ter than the 3-σ level and r = 0.03 at about 3-σ level.

On the other hand, delensing can be implemented if asmall, clean patch of the sky is observed with high sensi-tivity. In our analysis, if an experiment like QUIETBeaRcan achieve 1% foreground cleaning and can observe a par-ticularly clean patch of the sky, we forecast that delensingcan be successfully implemented; thus r = 0.01 can be de-

CMB polarization and inflation 13

Table 5. Ground-based and balloon-borne experiments (see Ta-ble 7 for specifications), including instrumental noise and fore-grounds, for flat cosmologies. τ = 0.164 has been taken to be thefiducial value unless explicitly stated otherwise. We have usedWMAP priors on cosmological parameters as explained in the

text.

case r ∆r ∆n

QUIET 0.01 0.015 0.029FG 1% 0.03 0.019 0.03

CR 0.1 0.035 0.03

QUIETBeaRDASI10%,dust1% 0.01 0.003 0.054

FG1% CR DL

PolarBeaR 0.01 0.012 0.018DASI50%,dust1% 0.03 0.015 0.018

FG1% CR 0.1 0.023 0.018

QUIET+PolarBeaR 0.01 0.006 0.015DASI50%,Pol1% 0.03 0.009 0.02

FG1% CR 0.1 0.017 0.02

SPIDER, τ = 0.1DASI50%,Pol1% 0.01 0.004 0.1

FG1% CR

tected at the ∼ 3-σ level if imposing the consistency relation(Table 5).

Note, however, that at small angular scales, extragalac-tic point source contamination – which we have neglected –may have an amplitude comparable to that of the lensingsignal, making delensing implementation even harder.

The specifications of the SPIDER experiment are nottoo dissimilar from those of the realistic satellite experimentconsidered above. The slightly different frequency coverageand noise level, the different beam size and the fact that theremaining fraction of the sky after the Galactic cut is 40%,change the forecast errors only slightly. For example for afiducial model with τ = 0.1, r = 0.01 can be detected at the2-σ level in the presence of noise and foregrounds. Thus, thesame considerations for the satellite experiments apply here.We report forecasts for SPIDER in Tables 5 and 6. Note thatthe errors on other parameters which are mostly constrainedby the temperature data at ℓ > 300 such as n, wb etc. arelarger than for our satellite experiment. This is because thebeam size of SPIDER is not optimized for this. The error onr for the CR case is only a factor of a few larger than for asatellite experiment for τ = 0.1. Such an experiment couldperform two flights, one in the northern hemisphere and onein the southern hemisphere, and thus cover a larger fractionof the sky. Assuming that the two maps can be accuratelycross-calibrated we find that the errors on r could be reducedscaling approximately like

√

fsky .

6 CONCLUSIONS

One issue that we have not discussed is that, while on thefull sky E and B modes are separable, this is no longerthe case when only small patches of the sky are ana-lyzed, as the boundaries of the patch generate mode mix-ing. The smaller the patch, the more important the effect

of the boundary is expected to be. In the absence of fore-grounds, Lewis, Challinor & Turok (2002) showed that forsimple patch geometries, E and B modes can still be sepa-rated. In particular, for circular patches larger than about5 degrees in radius, they find that the degradation due toimperfect E–B separation is negligible. However, for moregeneral patch shapes this may be optimistic. All the exper-iments we considered had patches comfortably larger thanthis limit, and therefore we expect that this effects shouldnot degrade our forecasts significantly.

From the above results we can deduce some general con-siderations which may help in planning, designing and opti-mizing future B-mode polarization experiments. Given therealistically achievable constraints on r, we can then forecastwhat can be learned about inflationary physics from theseexperiments. In our forecast, we have also neglected “realworld” effects that are experiment-specific such as inhomo-geneous/correlated noise, 1/f effects, etc. But these effectsare not expected to significantly alter our conclusions (PO-LARBeaR and QUIET collaborations, private comm.).

6.1 Guidance for B-mode polarization

experiments

Our assumptions about foregrounds contamination arebased on information coming from much lower or higherfrequencies than the CMB window, and in many cases fromobservations of patches of the sky. Since the properties of thepolarized foregrounds show large spatial variations acrossthe sky, the optimal “cosmological window” may dependstrongly on the area of the sky observed and may be dif-ferent for full sky or partial sky experiments. In addition,experiments working below ∼ 70 GHz will want to concen-trate in sky regions with low synchrotron contamination,while experiments at higher frequencies need to focus re-gions with low dust: one “recipe” may not “fit all”. Dueto our limited polarized foreground knowledge it is not yetpossible to make reliable predictions on which areas of thesky will be the cleanest and more suitable for primordialB-mode detection (or if the experiments considered here, byreducing their sky coverage, could be less contaminated byforegrounds). However forthcoming datasets (e.g., WMAPpolarized maps) combined with existing ones (e.g., Archeopsand Boomerang polarization maps) will enable one to do so.

The primordial B-mode signal on large scales (accessi-ble by space-based or balloon-borne experiments) is not con-taminated by lensing, but its amplitude is affected by theoptical depth to reionization τ . The lensing signal (domi-nant on scales ℓ >∼ 50) can be removed to high accuracy(delensing) in the absence of noise and foregrounds. Fore-ground contamination and noise in the foreground templatesare the limiting factor in constraining r and in the de-lensing implementation. For the considered realistic space-based experiments the residual noise (both in the maps andin foreground templates) hinders delensing implementation.Delensing may be used to improve r limits in partial skyground-based experiments: partial sky experiments can eas-ily achieve lower noise level than space-based ones and cantarget particularly clean areas of the sky, but accurate fore-ground templates are still needed to keep foregrounds con-tamination at or below the 1% level and to reduce templatenoise.

14 Verde, Peiris & Jimenez

A space-based or balloon-borne experiment can easilymeasure r = 10−3 (at the ∼3−sigma level) if foregroundscan be subtracted at 1% level, but foreground contaminationis ultimately the limiting factor for a detection of r ∼ 10−4.

From the ground, r = 0.01 can be detected at the 2-σlevel but only if foreground can be subtracted at the 1%level and if accurate foreground templates are available.

In order to have accurate and high resolution foregroundtemplates and for the foregrounds to be subtracted at thepercent level, a drastically improved knowledge of the am-plitude and the spectral and spatial dependence of polarizedforeground emission is needed. This can be achieved by ob-serving the polarized foreground emission on large areas ofthe sky, at high Galactic latitudes, with high resolution andhigh sensitivity at multiple wavelengths, both at higher andlower frequencies than the “CMB window”. The success offuture B-modes experiments will rely on these efforts. On-going and planned experiments such as EBEX, CLOVER,QUaD, Planck, BLASTPol, will play a crucial role in achiev-ing this goal.

If this can be achieved, in order to constrain r as bestas possible, a combination of approaches will most likely beneeded. A space-based or balloon-borne experiment can beoptimized to observe the low ℓ BB “bump”: it can have awide frequency coverage, but weight restrictions can be metby having less stringent requirements for noise and angu-lar resolution. Future advances to ballooning methods, suchas “stratellite”12 technology may make that avenue particu-larly attractive. A stratellite has a flight time of 18 months’duration, is stationed at 65,000 feet and is capable of car-rying a payload of up to 3000 lb. In addition the airshipis 100% reclaimable and the vehicle will be much cheaperto build and to run than a satellite. These properties maymake such an experiment achieve the sky coverage and in-tegration time of a satellite, retaining at the same time theadvantages of a balloon-borne experiment (e.g. lower costs,less weight restrictions, upgradable).

A satellite/balloon-borne experiment is nicely comple-mentary to a ground-based partial-sky experiment. Theground-based telescope can be optimized to implement de-lensing: it can observe a particularly clean patch of the sky;and can achieve low noise level by using large detectors ar-rays. Thus the noise and beam size requirements need tobe more stringent than for a full sky experiment tailored toaccess the reionization bump, and the frequency coveragemust be wide enough to still enable accurate (percent level)foreground subtraction.

6.2 Implications for Inflation

What can experiments with the capabilities illustratedhere tell us about inflationary physics? Figures 1 and 2show that, to produce a clearly detectable (>3-σ) tensorcomponent in any foreseeable CMB experiment, inflationmust necessarily involve large-field variations: ∆φ >∼ 1. AsEfstathiou & Mack (2005) point out, the relation in Eq. 11is so steep that, to probe models with small field variationswhere an effective field theory description is likely to be

12 Stratellite: http://www.sanswire.com

Figure 4. Otherwise identical to Fig. 1 except that the con-straints on ns and dns/d ln k are 2-σ limits from the CR case ofTable 3.

Figure 5. Otherwise identical to Fig. 2 except that the con-straints on ns and dns/d ln k are 2-σ limits from the CR case ofTable 3.

valid, one needs to be able to detect r 6 10−4. This is ex-ceedingly difficult in CMB experiments, given realistic po-larized foregrounds, detector noise achievable in the fore-seeable future, and the weak lensing contamination of theprimordial B-modes. Therefore, in the foreseeable future,it appears that we are limited to testing large-field mod-els of inflation. Figures 4 and 5 show what happens to the∆φ vs. r relation when the observational limits on ns anddns/d ln k at the level of the 2-σ limits from the CR caseof Table 3 are imposed. Observation of a significant ten-sor/scalar ratio may therefore be a signal that inflation isbeing driven by some physics in which the inflaton cannotbe described as a fundamental field captured in terms of alow energy effective field theory description such as Eq. 6.

CMB polarization and inflation 15

Note that the converse of this argument is not implied; thereare perfectly valid inflation models realized within the effec-tive field theory framework in which the inflaton is not afundamental field. Alternatively, as Linde (2005) points out,the effective field theory description of V (φ) becomes invalidonly for V (φ) > m4

Pl, not φ > mPl as inferred from Eq. 6.Chaotic inflation models, such as the simple λφ4 model (ex-cluded at about the 3-σ level by current cosmological con-straints), exploit this rationale; on the other hand, buildingrealistic particle-physics motivated chaotic inflation modelshas historically proved difficult. However, it has been shownrecently that chaotic inflation models can be realized in su-pergravity models if the potential has a shift symmetry (seeLinde (2005) and references therein for a detailed explana-tion). Thus it might be possible to construct chaotic inflationmodels motivated by realistic particle physics with ∆φ >∼ 1.Further, as Boyanovsky D., de Vega H. L. & Sanchez N. G.(2005) point out, inflation can be described as a series ex-pansion in terms of a new variable χ = φ/(

√NmPl), which

makes the effective field theory description valid up to r ∼ 1.In this case φ is not a fundamental field but a degree of free-dom in the theory.

It is therefore important to investigate the possi-bility of constructing particle-physics motivated inflation-ary models with ∆φ >∼ 1, since these are likely to bethe only models that could be probed by realistic CMBexperiments in the near future. Significant progress inthis direction has already been made; for example, ex-tranatural/pseudonatural inflation Hill & Leibovich (2002);Arkani-Hamed et al. (2003a,b), in which the inflaton is theextra component of a gauge field in a 5-dimensional theorycompactified in a circle, which predicts a significant produc-tion of a tensor component (r ∼ 0.11). Purely 4d theoriesappear to require more sophisticated structures in order toprotect the flatness of the potential from excessive radiativecorrections (Kim et al. 2005; Arkani-Hamed et al. 2003b),and in general do not predict significant gravity-wave pro-duction. However they still appear to preserve the predictionof a significant deviation from scale invariance as a signatureof inflation.

Conversely, if tensor modes are not detected in these ex-periments, we can exclude large-field models as a mechanismfor inflation. Then we would need to examine the remain-ing possibilities by investigating constraints on the deviationfrom scale-invariance of the scalar spectral index (a blue orred index discriminates between classes of inflationary mod-els), and also from its running. While this paper focuseson the implications of a primordial B-mode measurementfrom foreseeable CMB experiments for inflation, a negligi-ble primordial tensor component is also a prediction of theEkpyrotic/Cyclic scenarios (Khoury et al. 2002, 2001).

6.3 “Bang for your bucks”

To conclude, in Figure 6 we summarize the minimum ex-perimental set up and foreground knowledge that is neededto measure a given r value at >∼ 3–σ level. We also indicatewhether the signal comes from the reionization “bump” re-gion (affected by τ ) or the “first peak” region (affected bylensing contamination) and if delensing can be used to im-prove signal to noise. Finally we label the experiments withletters a for ground based, b for balloon-borne and c for

Figure 6. Superimposed upon the Monte-Carlo points of Fig. 4are the minimum values of r that can be measured at the 3–σlevel for different experimental setups and foreground knowledge.The legend on the right indicates whether the signal comes fromthe “peak” (ℓ > 30) region or the “bump” (ℓ < 30) region; ifthe experiment is ground-based (a), balloon-borne (b) or satel-lite (c); and how expensive such an experiment may be. The topdashed line shows the current upper limit. Single ground-basedexperiments can reach r = 0.1 if foreground subtraction can bedone at the 1% level. The signal in this case would come from thepeak region and the noise characteristic would not allow delens-ing. If two different experiments such as QUIET and PolarBeaRcan i) overlap their survey region and ii)select a sky patch withrealistic synchrotron contamination (50% of DASI upper limit)and low polarized dust emission (1% of Archeops limit) and iii)clean foregrounds to the 1% level, the wider frequency coverageand the lower noise means that r = 0.03 can be reached. Also,in this case the signal comes from the peak region. r = 0.01 canbe reached in two different ways. The technologies of QUIET andPolarBeaR could be combined: QUIET noise properties can beextended to also include PolarBeaR channels, a smaller regionof the sky is scanned so that the noise per pixel can be reducedand delensing can be implemented; and the sky region chosenis particularly clean (dust: 1% of Archeops limit; synchrotron:10% of DASI limit). The signal would come from the “peak” andthe measurement would rely on delensing. Alternatively a longduration balloon flight with the specifications of SPIDER wouldmeasure the same r but accessing the reionization bump regionand without relying on delensing. r = 10−3 can be achieved onlywith a satellite mission. Finally, foreground contamination, evenif at the 1% level, is the limiting factor for accessing r = 10−4

with an ideal, space based, noiseless experiment.

satellite. These could also be interpreted as price-tags fromleast expensive (a) to most expensive (c). There is a clearcorrelation between cost of the experiment and models thatcan be accessed by it.

16 Verde, Peiris & Jimenez

ACKNOWLEDGMENTS

We thank Mark Devlin, George Efstathiou, Chao-LinKuo, Stephan Meyer, Lyman Page, John Ruhl, DorotheaSamtleben, Kendrick Smith, David Spergel, Huan Tran, BenWandelt, and Bruce Winstein for useful conversations, andEiichiro Komatsu and Adrian Lee for comments on an earlyversion of the manuscript. LV is supported by NASA grantADP03-0000-0092 and ADP04-0000-0093. HVP is supportedby NASA through Hubble Fellowship grant #HF-01177.01-A awarded by the Space Telescope Science Institute, whichis operated by the Association of Universities for Researchin Astronomy, Inc., for NASA, under contract NAS 5-26555.She acknowledges the hospitality of the Kavli Institute forParticle Astrophysics and Cosmology at Stanford Univer-sity, where part of this work was carried out. The researchof RJ is partially supported by NSF grant AST-0206031.

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APPENDIX

18 Verde, Peiris & Jimenez

Table 6. Constraints on all parameters. IDEAL denotes an ideal experiment(no foregrounds, cosmic variance dominated). SAT denotesa space-based experiment. CR means that the consistency relation for nT , nt = −r/8 has been imposed. FLT means that flatness hasbeen imposed, this does not affect the errors on the parameters relevant to inflation. FG denotes that galactic foregrounds has beenconsidered. FG1% means that their residual contamination is reduced to 1% while FG10% means that foreground contamination hasbeen reduced to 10%. The experimental specifications used for obtaining these constraints are reported in Table 7. The error on the

amplitude does not include possible calibration errors which are likely to dominate. These forecasts have been computed for a fiducialvalue of τ = 0.164 except where explicitly specified. As the BB signal on large scales is boosted by τ roughly as τ2, lower values of τ willdegrade the significance of these forecasts. For fiducial τ one sigma below the best fit value the detectability of r = 0.01 (r = 0.03) fromℓ < 20 is equivalent to the detectability of r = 0.03 (r = 0.1) in the best fit model. This consideration affects only large sky coverageexperiments which probe the “reionization bump” in the BB spectrum at ℓ < 10 and does not affect the smaller scale ground-basedexperiments.

r case ∆r ∆n ∆nt ∆dn/d ln k ∆Z ∆ωb ∆ωc ∆h ∆ΩK ∆A

0.01 0.001 0.0017 0.056 0.0034 0.0033 6 × 10−5 0.00027 0.003 0.0006 0.0040.03 IDEAL 0.0028 0.0017 0.047 0.0036 0.0034 6.4 × 10−5 0.00026 0.0033 0.0006 0.0040.1 0.0063 0.0018 0.035 0.0035 0.0036 6 × 10−5 0.00020 0.0023 0.0006 0.004

0.01 0.00045 0.0017 – 0.0036 0.0032 6.1 × 10−5 0.00025 0.003 0.0006 0.00380.03 IDEAL, 0.00074 0.0017 – 0.0036 0.0032 6.4 × 10−5 0.00026 0.0032 0.00060 0.00380.1 CR 0.0015 0.0017 – 0.0036 0.0032 6.4 × 10−5 0.00026 0.0024 0.0006 0.0038

0.01 IDEAL 0.000021 0.0021 0.0019 0.0038 0.0038 6.5 × 10−5 0.00072 0.0057 0.0006 0.00480.03 DL 0.000063 0.0021 0.0019 0.0038 0.0038 6.5 × 10−5 0.00072 0.0057 0.0006 0.0049

0.01 IDEAL 0.000016 0.0021 – 0.0038 0.0038 6.5 × 10−5 0.00072 0.0057 0.0006 0.00480.03 DL CR 0.000049 0.0021 – 0.0038 0.0038 6.4 × 10−5 0.00072 0.0057 0.0006 0.0049

0.01 0.0030 0.0023 0.098 0.0046 0.0053 8.4 × 10−5 0.00053 0.0055 0.00078 0.00660.03 SAT 0.0048 0.0023 0.069 0.0046 0.0053 8.4 × 10−5 0.00054 0.0056 0.00080 0.00660.1 0.010 0.0023 0.056 0.0046 0.0055 8.4 × 10−5 0.00055 0.0058 0.00082 0.0068

0.01 SAT 0.0030 0.0023 0.096 0.0046 0.0049 8.3 × 10−5 0.00050 0.0023 – 0.00620.03 0.0047 0.0023 0.068 0.0046 0.0050 8.3 × 10−5 0.00051 0.0023 – 0.00620.1 FLT 0.010 0.0023 0.054 0.0046 0.0050 8.3 × 10−5 0.00052 0.0023 – 0.0062

0.01 0.0011 0.0023 – 0.0046 0.0052 8.3 × 10−5 0.00051 0.0054 0.00078 0.00650.03 SAT 0.0017 0.0023 – 0.0046 0.0050 8.3 × 10−5 0.00051 0.0054 0.00078 0.00630.1 CR 0.0028 0.0023 – 0.0046 0.0049 8.2 × 10−5 0.00051 0.0054 0.00080 0.0062

0.01 SAT 0.0031 0.0023 0.098 0.0046 0.0049 8.3 × 10−5 0.0005 0.0024 – 0.0063FG1% FLT

0.01 +CR 0.001 0.0022 – 0.0046 0.0049 8.3 × 10−5 0.0005 0.0023 – 0.0062

SAT,0.01 DASI50%, τ = 0.1 0.0030 0.0021 0.1 0.0044 0.0040 7.6 × 10−5 0.00047 0.0021 – 0.0045

FG1% FLT0.01 +CR 0.0012 0.0020 – 0.0043 0.040 7.6 × 10−5 0.00046 0.0020 – 0.0045

0.0001 SAT FLT 0.00019 0.0023 0.43 0.0046 0.005 8.3 × 10−5 0.00051 0.0023 – 0.00630.001 FG1% DASI50% 0.0013 0.0023 0.31 0.0046 0.005 8.3 × 10−5 0.0005 0.0023 – 0.0063

0.0001 SAT FLT CR 0.00012 0.0023 – 0.0046 0.005 8.3 × 10−5 0.0005 0.0023 – 0.00630.001 FG1% DASI50% 0.0003 0.0023 – 0.0046 0.005 8.3 × 10−5 0.0005 0.0023 – 0.0063

0.01 SAT 0.0035 0.0023 0.11 0.0046 0.005 8.3 × 10−5 0.00052 0.0024 – 0.0063FG10% FLT

0.01 +CR 0.0013 0.0023 – 0.0046 0.0049 8.3 × 10−5 0.0005 0.0023 – 0.0063

0.001 SAT FLT 0.00051 0.0023 0.13 0.0046 0.0050 8.3 × 10−5 0.00051 0.0023 – 0.0063FG 10% DASI50%

0.001 SAT FLT CR 0.00042 0.0023 – 0.0046 0.0050 8.3 × 10−5 0.00051 0.0023 – 0.0063FG 10% DASI50%

SAT NO NOISE0.0001 FG1% FLT 0.00007 0.0019 0.17 0.0037 0.005 7.4 × 10−5 0.0022 0.0008 – 0.005

DASI 50%0.0001 +CR 0.000057 0.0019 – 0.0037 0.005 7.4 × 10−5 0.0022 0.0008 – 0.005

CMB polarization and inflation 19

r case ∆r ∆n ∆nt ∆dn/d ln k ∆Z ∆ωb ∆ωc ∆h ∆ΩK ∆A

0.01 QUIET 0.09 0.029 4.2 0.047 0.06 0.0007 0.005 0.026 – 0.0830.03 FG1% 0.31 0.03 4. 0.048 0.062 0.0007 0.005 0.025 – 0.0830.1 FLT 0.32 0.03 1.3 0.48 0.062 0.0007 0.005 0.025 – 0.083

0.01 QUIET 0.015 0.029 – 0.045 0.061 0.00071 0.005 0.024 – 0.080.03 FG1% FLT 0.019 0.03 – 0.045 0.062 0.0072 0.005 0.025 – 0.080.1 CR 0.035 0.03 – 0.046 0.062 0.00072 0.005 0.024 – 0.08

0.01 QUIET 0.13 0.03 6.0 0.047 0.063 0.0007 0.005 0.023 – 0.080.03 FG 10% 0.32 0.032 5.1 0.05 0.063 0.0007 0.05 0.025 – 0.080.1 FLT 0.35 0.032 1.4 0.05 0.063 0.0007 0.005 0.024 – 0.08

0.01 QUIET 0.019 0.03 – 0.047 0.06 0.0007 0.005 0.023 – 0.080.03 FG10%,FLT 0.024 0.047 – 0.047 0.06 0.0007 0.005 0.023 – 0.080.1 CR 0.037 0.024 – 0.046 0.06 0.0007 0.005 0.023 – 0.08

0.01 PolarBeaR 0.07 0.018 3.7 0.037 0.049 0.0006 0.0043 0.018 – 0.060.03 DASI50%, Dust1% 0.1 0.018 2.1 0.037 0.051 0.00062 0.0044 0.019 – 0.060.1 FG1% FLT 0.1 0.018 0.59 0.037 0.052 0.00062 0.0044 0.019 – 0.06

0.01 PolarBeaR 0.012 0.018 – 0.037 0.051 0.00061 0.0043 0.019 – 0.060.03 DASI50%, dust1% 0.015 0.018 – 0.037 0.051 0.00061 0.0043 0.019 – 0.060.1 FG1% FLT CR 0.023 0.018 – 0.037 0.052 0.00061 0.0043 0.019 – 0.06

0.01 QUIET+PolarBeaR 0.036 0.017 2.0 0.033 0.058 0.00066 0.0046 0.02 – 0.0670.03 DASI50%dust1% 0.08 0.02 1.7 0.039 0.063 0.00067 0.0049 0.021 – 0.0670.1 FLT FG 1% 0.08 0.02 0.47 0.039 0.06 0.00064 0.0047 0.02 – 0.067

0.01 QUIET+PolarBeaR 0.006 0.015 – 0.03 0.058 0.00066 0.0046 0.02 – 0.0670.03 DASI50%dust1% 0.009 0.02 – 0.039 0.063 0.00067 0.0048 0.021 – 0.0670.1 FLT FG 1% CR 0.017 0.02 – 0.039 0.06 0.00064 0.0046 0.02 – 0.067

QUIETBeaR0.01 DASI10%, dust1% 0.017 0.054 1.0 0.094 0.15 0.0016 0.01 0.05 – 0.19

FG 1% FLT0.01 +CR 0.003 0.054 – 0.093 0.15 0.0016 0.01 0.05 – 0.19

SPIDER, τ = 0.10.01 DASI50% 0.02 0.1 0.44 0.06 0.007 0.002 0.01 0.04 – 0.1

FG 1% FLT0.01 +CR 0.004 0.1 – 0.06 0.007 0.002 0.009 0.03 – 0.09

20 Verde, Peiris & Jimenez

Table 7. Experimental specifications used for computing cosmological constraints.

frequency noise/beam beam FWHM sky coverage(GHz) (µK) (′) (fsky or sq. deg.)

SAT 30, 50, 70, 100, 200 2.2 per channel 8 fsky=0.8

QUIET 40 0.43 23 4 × 40090 0.78 10

PolarBeaR 90 1.6 6.7 500150 2.4 4.0220 11.3 2.7

QUIET+PolarBeaR 40 0.43 23 400

90 0.78 1090 1.6 6.7150 2.4 4.0220 11.3 2.7

QUIETBeaR 40 0.1 23 17090 0.18 1090 0.18 10150 0.18 10220 0.18 10

SPIDER 40 0.74 145 fsky=0.484 0.36 6992 0.36 63100 0.36 58145 0.58 40220 1.6 26